The Critical Group of a Line Graph

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph.

• The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph.

• The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices.

• The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups.

The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion.     

Single-step kinase inhibitor screening using a peptide-modified gold nanoparticle platform

Two complementary formats for kinase inhibitor screening are presented in which peptide-modified gold nanoparticles are enzymatically phosphorylated and rapidly aggregate on a surface or in solution by action of phosphospecific antibodies. The simple and rapid colourimetric response of the assays makes them an attractive approach for drug-screening applications.     

Edge integrity of nearest neighbor graphs and separator theorems

We give upper bounds for the edge integrity and the vertex integrity of $k$-type nearest neighbor graphs in ${\mathbb{R}}^d$. For $n$ the order of a graph, examples are constructed to show that the bounds are best possible in the parameters $n$ and $k$. The results follow from decompositions that are shown for certain graphs that possess small edge or vertex separators. A vertex separator theorem for nearest neighbor graphs is known, and an edge separator theorem is shown. We also apply the decompositions to obtain upper bounds for the edge integrity of planar graphs, graphs of fixed genus, and trees.     

Orientational order and molecular parameters of two nematogenic mixtures from X-ray studies: confirmation of electro-optical results

This study entails the report of X-ray studies conducted on two nematogenic mixtures (code names ZLI 1221 and ZLI 1291) containing identical moieties: phenyl cyclohexane, biphenyl cyclohexane and cyclohexyl benzoate, but exhibiting different mesogenic ranges and properties. X-ray studies have been undertaken with a view to investigate the difference in the mesogenic behaviour of the said mixtures at the molecular level. The average apparent molecular length ‘l?’ and intermolecular distance ‘D’ of the two mixtures and their variations with temperature have been determined from X-ray data. The orientational order parameters ?P ₂? and ?P ₄? and their thermal variations have also been obtained. Comparison has been made with the ?P ₂? values obtained from X-ray study with those obtained from optical study. The average apparent molecular lengths of the two mixtures corroborate the findings regarding the relative molecular weights of the two samples obtained from optical and dielectric data.     

Stability of Self-Assembled Polymeric Micelles in Serum

The stability of polymeric nanoparticles in serum is critical to their use in drug delivery where dilution after intravenous injection often results in nanoparticle disassembly and drug unloading; however, few investigate this in biologically relevant media. To gain greater insight into nanoparticle stability in blood, the stability of self-assembled polymeric micelles of poly(d,l-lactide-co-2-methyl-2-carboxytrimethylene carbonate)-g-poly(ethylene glycol), P(LA-co-TMCC)-g-PEG, were tested in both serum and individual serum protein solutions. By encapsulating F?rster resonance energy transfer pairs and following their release by fluorescence, these micelles demonstrated excellent thermodynamic and kinetic stability in the presence of serum. Further analyses by fast protein liquid chromatography and dynamic light scattering confirmed these data. Moreover, these micelles are compatible with red blood cells, as shown by a hemolysis assay. The stability and compatibility demonstrated in blood suggest that these micelles may be stable in vivo, which is critical for intravenous drug delivery applications. This comprehensive approach to understanding micelle stability and compatibility is broadly applicable.     

Lactide polymerization co‐initiated by carbohydrate esters and pyranoses

The polymerization of [S]‐lactide was accomplished using an initiating system comprising an alkyl zinc complex and a series of well defined carbohydrate co‐initiators derived from D ‐glucose, D ‐xylose, and 2‐deoxy‐D ‐ribose. The monosaccharide co‐initiators were aldonate esters and pyranoses, they were all prepared in high yield and had only a single alcohol co‐initiating group; the remaining carbohydrate hydroxyl functionalities were protected as acetyl, benzyl ether and isopropylidene acetal groups. The polymerizations were all well controlled, illustrated by the linear increase in poly(S‐lactide) M_n with percentage conversion of lactide, the increase in poly(S‐lactide) M_n with [lactide]₀‐[lactide]_t/[co‐initiator] and the narrow polydispersity indices of the polylactides. Thus, the novel initiating systems were used to produce poly(S‐lactides) end functionalized with a variety of different aldonate ester and pyranose groups and with degrees of polymerization from 10 to 250. The polyesters were fully characterized, including by NMR spectroscopy, size exclusion chromatography (SEC), matrix‐assisted laser deposorption/ionization (MALDI) mass spectrometry and by static water contact angle measurements. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 4350–4362, 2008